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Bayesian Filtering for Location Estimation

Bayesian Filtering for Location Estimation. D. Fox, J. Hightower, L. Liao, D. Schulz, and G. Borriello. Presented by: Honggang Zhang. Outline. Basic idea of Bayes filters Several types of Bayes filters Some applications. System state dynamics. Observation dynamics.

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Bayesian Filtering for Location Estimation

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  1. Bayesian Filtering for Location Estimation D. Fox, J. Hightower, L. Liao, D. Schulz, and G. Borriello Presented by: Honggang Zhang

  2. Outline • Basic idea of Bayes filters • Several types of Bayes filters • Some applications

  3. System state dynamics Observation dynamics We are interested in: Belief or posterior density Bayes Filters Estimating system state from noisy observations

  4. Recall “law of total probability” and “Bayes’ rule” From above, constructing two steps of Bayes Filters Predict: Update:

  5. Assumptions: Markov Process Predict: Update:

  6. Bayes Filter How to use it? What else to know? Motion Model Perceptual Model Start from:

  7. Step 0: initialization Step 1: updating Example 1

  8. Step 3: updating Step 4: predicting Step 2: predicting Example 1 (continue)

  9. Several types of Bayes filters • They differs in how to represent probability densities • Kalman filter • Multihypothesis filter • Grid-based approach • Topological approach • Particle filter

  10. Recall general problem Assumptions of Kalman Filter: Belief of Kalman Filter is actually a unimodal Gaussian Advantage: computational efficiency Disadvantage: assumptions too restrictive Kalman Filter

  11. Multi-hypothesis Tracking • Belief is a mixture of Gaussian • Tracking each Gaussian hypothesis using a Kalman filter • Deciding weights on the basis of how well the hypothesis predict the sensor measurements • Advantage: • can represent multimodal Gaussian • Disadvantage: • Computationally expensive • Difficult to decide on hypotheses

  12. Grid-based Approaches • Using discrete, piecewise constant representations of the belief • Tessellate the environment into small patches, with each patch containing the belief of object in it • Advantage: • Able to represent arbitrary distributions over the discrete state space • Disadvantage • Computational and space complexity required to keep the position grid in memory and update it

  13. Topological approaches • A graph representing the state space • node representing object’s location (e.g. a room) • edge representing the connectivity (e.g. hallway) • Advantage • Efficiency, because state space is small • Disadvantage • Coarseness of representation

  14. Particle filters • Also known as Sequential Monte Carlo Methods • Representing belief by sets of samples or particles • are nonnegative weights called importance factors • Updating procedure is sequential importance sampling with re-sampling

  15. Step 0: initialization Each particle has the same weight Step 1: updating weights. Weights are proportional to p(z|x) Example 2: Particle Filter

  16. Step 3: updating weights. Weights are proportional to p(z|x) Step 4: predicting. Predict the new locations of particles. Step 2: predicting. Predict the new locations of particles. Example 2: Particle Filter Particles are more concentrated in the region where the person is more likely to be

  17. Compare Particle Filter with Bayes Filter with Known Distribution Updating Example 1 Example 2 Predicting Example 1 Example 2

  18. Comments on Particle Filters • Advantage: • Able to represent arbitrary density • Converging to true posterior even for non-Gaussian and nonlinear system • Efficient in the sense that particles tend to focus on regions with high probability • Disadvantage • Worst-case complexity grows exponentially in the dimensions

  19. Comparison + : good; 0 : neutral; - : weak

  20. Example Applications • Particle Filters (unconstrained) • Particle Filters (constrained) • Combination of Particle Filters and Kalman Filters

  21. Sensors • Ultra sound and infrared Sensors: • Less accurate but certain with identification • Laser range finder • Accurate but anonymous

  22. Example Indoor Environment Red circles: ultra-sound ID sensors Blue squares: infrared ID sensors

  23. Using Particle Filters (unconstrained) • Due to high noise level of ultrasound and infrared sensors, we use particle filters • Whenever detect the person, updating particles

  24. Using Particle Filters (unconstrained) Another Example

  25. Using Particle Filters (unconstrained) Another Example

  26. Using Particle Filters (constrained) • A more efficient way to use particle filters • constraining the state space to locations on a Voronoi graph (a structure similar to a skeleton of an environment’s free space)

  27. Combine Particle and Kalman Filters To Solve Data Association Problem Laser range finder Data Association Problem In area 5 and 6, resolving ambiguity, but need additional hypotheses In area 3 and 4, identities of A and B are known Area covered by ID sensors

  28. Combine Particle and Kalman Filters To Solve Data Association Problem • Track individual people using Kalman filters (using laser range data) • A particle filter maintains multiple hypothesis wrt identities of people

  29. Conclusion • “The Location Stack”: a general framework with publicly available implementation • Probabilistic techniques have tremendous potential for inference problems Questions?

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